One-quasihomomorphisms from the integers into symmetric matrices

TL;DR

This paper investigates functions from integers to symmetric matrices that nearly preserve addition, showing they are close to true homomorphisms, thus addressing a specific case of a problem in algebraic stability.

Contribution

It proves that 1-quasihomomorphisms from integers to symmetric matrices are within distance 2 of genuine homomorphisms, solving a particular case of Kazhdan and Ziegler's problem.

Findings

Any 1-quasihomomorphism has distance at most 2 from a true homomorphism.

The result applies over any field of characteristic zero.

Addresses a special case of a problem by Kazhdan and Ziegler.

Abstract

A function $f$ from $\mathbb{Z}$ to the symmetric matrices over an arbitrary field $K$ of characteristic $0$ is a $1$-quasihomomorphism if the matrix $f(x+y) - f(x) - f(y)$ has rank at most $1$ for all $x,y \in \mathbb{Z}$. We show that any such $1$-quasihomomorphism has distance at most $2$ from an actual group homomorphism. This gives a positive answer to a special case of a problem posed by Kazhdan and Ziegler.